# Category Archives: Mathematics

## Introduction to Wasserstein metric (earth mover’s distance)

Table of Contents 1. the cost of moving one particle 2. The cost of moving a finite collection of particles of total mass 1. 3. coupling 4. composing 5. definition of the Wasserstein distance 6. Moving a collection of particles … Continue reading

## Intuitions on problems from Elements of Information Theory Chapter 2

Table of Contents 1. meaning of conditional mutual information 2. comments on some selected problems from Chapter 2 2.1. Problem 2.6 2.2. Problem 2.14 2.3. Problem 2.23 2.4. Problem 2.27 2.5. Problem 2.41 This post is not about sharing solutions … Continue reading

## Doob-Dynkin Lemma for probability space

Table of Contents 1. Proof 1 2. Proof 2 3. Proof 3 4. Uniqueness of H 5. applications Let f, g be measurable functions from a probability space to measurable spaces and respectively. Consider the following three conditions: (1) (in … Continue reading

## Proofs of Doob-Dynkin Lemma

Table of Contents 1. uniqueness of H 2. counterexamples 3. proof when g has a countable range, with nice Y 4. proof when Y is the Cantor space 5. proof when Y is the unit interval 6. proof for standard … Continue reading

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## Small introduction to Hausdorff distance

Table of Contents 1. motivation and intuition 2. coming up with the two definitions of the Hausdorff distance 3. more exercises Update: I realized that parts of this post focus on unimportant small details. Sorry for that, you may skip … Continue reading

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## mod 0 equivalence of sub-sigma-algebras and gotchas

Table of Contents 1. closure of a sub-sigma-algebra 2. closed sub sigma algebra 3. mod zero equivalence of sub sigma algebras When we are building some theory involving measures, there are cases where we should ignore null sets (sets of … Continue reading